Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics

Abstract

In the frame of a generic language extended from the polarization theory-comprising the notions of Poincaré vectors, Poincaré sphere, and P-spheres-a geometric approach to Lorentz transformations alternative to the Minkowskian one is given. Unlike the four-dimensional Minkowskian approach, this new approach operates in the three-dimensional space of Poincaré vectors.

Quantum Polarimetry

“…Instead, the product of two boosts becomes the composition of a boost and a rotation. This extra rotation is present in many physical situations (Malykin, 2006;Tudor, 2018), such as through the Thomas precession in special relativity (Thomas, 1926) and the Wigner rotation in mathematical physics (Wigner, 1939). This effect has indeed been investigated for light's polarization degrees of freedom (Monzón and Sánchez-soto, 2001;Monzón and Sánchez-Soto, 1999a,b;Vigoureux, 1992;Vigoureux and Grossel, 1993), with the fascinating mathematical equivalence between indices of refraction for polarized light traveling through planar media and relative velocities of inertial frames in special relativity (Vigoureux, 1992).…”

“…Polarimetry and its relative ellipsometry (Azzam, 2011) do just that: they characterize substances by the unique changes they impart on light's polarization degrees of freedom (Collett, 1992). Highly precise measurements of polarization and its changes, i.e., polarimetry, have found applications in photonics (Yoon et al, 2020), bioimaging (Dulk et al, 1994;Firdous and Anwar, 2016;Ghosh et al, 2009;Tuchin, 2016), oceanography (Voss and Fry, 1984), remote sensing (Tyo et al, 2006), astronomy (Dulk et al, 1994;Tinbergen, 2005), and beyond.…”

On the correct mathematical description of polarization dependent impairments in Optical Communications

“…The main purpose of this paper is to show that the mathematical formulation that correctly describes polarization issues in optical communication is the spinor irreducible representation theory of the extended (as opposed to restricted) Lorentz Group. Although the idea of the Lorentz Group describing polarization in Optics is well known, [4]- [6] , it has never, to the best of my knowledge, been considered relevant to the field of optical communications. In this paper it will be analyzed why this happened, it will be proven that not taking into account these ideas is a serious mistake and it will be shown that polarization dependent (combined PMD-PDL) impairments can be easily reduced once the full implications of this mathematical formulation are understood.…”

“…[(ωΩ r ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ + j ωΩ i ⃗⃗⃗⃗⃗⃗⃗ ). σ ⃗ ⃗ ] and the Lorentz transformation can be obtained from the traceless 2x2 matrix (3) by the exponential map as shown in (4). From this point of view, the combined PMD and PDL vector generates a restricted Lorentz transformation of the output polarization states and, if we choose to use Jones vectors to describe polarization states, we can regard these as a proper (restricted) Lorentz transformations of spin ½ particles [9]- [12].…”

On the correct mathematical description of polarization dependent impairments in Optical Communications

“…Since there is a close relationship between the proper Lorentz group, SO + (1,3), and SL(2,C) (SL(2,C) is the double cover of SO + (1,3)) this seems to be just a mathematical curiosity devoid of any practical interest. Possibly this is the reason why, although this relationship is known in Optics [4]- [6], it has never attracted any attention in the field of optical communications. In the following sections it will be show that this was a mistake because combined PMD-PDL induced impairments can be easily alleviated (the PDL part can be cancelled) when this fact and its precise mathematical formulation are taken into account.…”